## On the existence theorems of Kantorovich, Miranda and Borsuk

Götz Alefeld, Andreas Frommer, Gerhard Heindl, and Jan Mayer

### Abstract

The theorems of Kantorovich, Miranda and Borsuk all give conditions on the existence of a zero of a nonlinear mapping. In this paper we are concerned with relations between these theorems in terms of generality in the case that the mapping is finite-dimensional. To this purpose we formulate a generalization of Miranda's theorem, holding for arbitrary norms instead of just the $l_{\infty}$-norm. As our main results we then prove that the Kantorovich theorem reduces to a special case of this generalized Miranda theorem as well as to a special case of Borsuk's theorem. Moreover, it turns out that, essentially, the Miranda theorems are themselves special cases of Borsuk's theorem.

Full Text (PDF) [132 KB], BibTeX

### Key words

nonlinear equations, existence theorems, fixed points, Newton-Kantorovich theorem, Miranda theorem, Borsuk theorem.

### AMS subject classifications

47H10, 47J05, 65H10.

< Back